This question illustrates the following theorem.If one player of a 2-person zero-sum game employs a fixed strategy, then the opponent has an optimal counter strategy that is pure. In other words, if Player I knows that Player II is using a particular mixed strategy y, then Player I can maximize their expected payoff by using a pure strategy, and vice versa.

a. Work out the following.i. Find the expected payoff, E(x,y), to Player I if Player Iuses mixed strategy (x1, 1-x1), 0 ≤ x1 ≤ 1, and Player II usesmixed strategy (y1, 1-y1), 0 ≤ y1 ≤ 1, in the 2-person zerosumgame with the following payoff matrix.

6 , 1−2 , 3

(this is a 2x2 matrix)

ii. Now, think of y1 is being fixed to y1=0.3, find the value(s)of x1 that maximizes E(x,y) for fixed y1. Make your casethat, in general, if Player I can use a pure strategy to getthe maximum expected payoff, given that Player I knowsPlayer II’s strategy.

b. Suppose that Player II knew that Player I is to use a mixedstrategy (0.3, 0.1, 0.6) in the game with the following payoffmatrix. Work out the three expected payoffs if Player II uses apure strategy and hence deduce the best strategy for Player II.